Review of complex numbers Z2=R2e2 Z1 Z2 R1 R2e1+82) 01+02 21=R1e01 z=Reia R $82=Re¹(6+2x/8) The complex conjugate of z = Reie = a + bi is z=Rea- bi, which is the reflection of z across the real axis. Do not use AI, I need real solution, attach required graph and code wherever needed. For reference I have attached the image, but if you need any reference then check out the book by Churchill only. C8=e²xi/8 Note that |z2=zz Rei Re-i = R² e° = R² |z|= √√√zz=√√√a² + b² = R. Explore the connections between potential theory and complex analysis. 1. Green's Functions in Complex Domains: ⚫ a. Define the Green's function for a domain DCC with a pole at $ € D. b. Construct the Green's function for the unit disk ID and verify its properties. 2. Poisson Kernel and Harmonic Functions: ⚫ a. Derive the Poisson kernel for the upper half-plane H = {z € C | I(2) > 0}. ⚫ b. Use the Poisson kernel to solve the Dirichlet problem for HI with prescribed boundary values on the real axis. 3. Harmonic Measure and Conformal Mappings: ⚫a. Define the harmonic measure w(z, E, D) for a subset ECOD and a point z € D. ⚫ b. Show how conformal mappings preserve harmonic measures and use this property to compute harmonic measures in transformed domains. 4. Capacity and Extremal Problems: ⚫ a. Define the conformal capacity of a compact set KCC. ⚫ b. Solve an extremal problem to find the conformal capacity of a circle of radius r centered at the origin. 5. Potential Theory and Analytic Functions: ⚫ a. Explain the relationship between subharmonic functions and analytic functions. b. Prove that the logarithm of the modulus of a non-constant analytic function is a subharmonic function.